Title:	Double Machine Learning Algorithms
Version:	1.0.2
Description:	Implementation of double machine learning (DML) algorithms in R, based on Emmenegger and Buehlmann (2021) "Regularizing Double Machine Learning in Partially Linear Endogenous Models" <doi:10.48550/arXiv.2101.12525> and Emmenegger and Buehlmann (2021) <doi:10.48550/arXiv.2108.13657> "Double Machine Learning for Partially Linear Mixed-Effects Models with Repeated Measurements". First part: our goal is to perform inference for the linear parameter in partially linear models with confounding variables. The standard DML estimator of the linear parameter has a two-stage least squares interpretation, which can lead to a large variance and overwide confidence intervals. We apply regularization to reduce the variance of the estimator, which produces narrower confidence intervals that are approximately valid. Nuisance terms can be flexibly estimated with machine learning algorithms. Second part: our goal is to estimate and perform inference for the linear coefficient in a partially linear mixed-effects model with DML. Machine learning algorithms allows us to incorporate more complex interaction structures and high-dimensional variables.
License:	GPL (≥ 3)
URL:	https://gitlab.math.ethz.ch/ecorinne/dmlalg.git
Encoding:	UTF-8
Depends:	R (≥ 4.0.0), stats
Suggests:	testthat (≥ 3.0.0)
Imports:	glmnet, lme4, matrixcalc, methods, splines, randomForest
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2022-02-03 12:20:14 UTC; Corinne
Author:	Corinne Emmenegger [aut, cre], Peter Buehlmann [ths]
Maintainer:	Corinne Emmenegger <emmenegger@stat.math.ethz.ch>
Repository:	CRAN
Date/Publication:	2022-02-03 12:40:02 UTC

Accessing the coefficients of regsdml fits

Description

This is a method for the class regsdml. It returns the estimated coefficients from objects of class regsdml, which typically result from a function call to regsdml.

Usage

## S3 method for class 'regsdml'
coef(object,
    print_regsDML = NULL,
    print_safety = NULL,
    print_DML = NULL,
    print_regDML = NULL,
    print_regDML_all_gamma = !is.null(parm),
    parm = NULL,
    print_gamma = FALSE, ...)

Arguments

Value

Coefficients of the methods regsDML, the safety device, DML, regDML with the optimal choice of \gamma (including the factor a_N), and regDML with prespecified \gamma-values are returned by setting the respective arguments. It is possible to return the respective gamma-values.

If none of the printing arguments are set, only the results of regsDML are returned if they are available. If they are not available and none of the printing arguments are set, the results from all available methods are returned. If print_regsDML = FALSE, only the results from those methods are returned that are explicitly specified by the printing arguments.

See Also

regsdml, summary.regsdml, confint.regsdml, vcov.regsdml print.regsdml

Examples

## See example(regsdml) for examples

Confidence Intervals for coefficient estimates of mmdml fits

Description

This is a method for the class mmdml. It computes two-sided confidence intervals for testing the two-sided component-wise null hypotheses H_0: \beta_j = 0 with the (approximate) asymptotic Gaussian distribution of the coefficient estimator. The method can be applied to objects of class mmdml that typically result from a function call to mmdml.

Usage

## S3 method for class 'mmdml'
confint(object, parm = NULL, level = 0.95, ...)

Arguments

Value

A matrix with columns giving the lower and upper confidence limits for each entry of \beta_0. The columns are labelled as These will be labelled as (1-level)/2% and 1 - (1-level)/2%, by default 2.5% and 97.5%.

See Also

mmdml

Examples

## See example(mmdml) for examples

Confidence Intervals for coefficient estimates of regsdml fits

Description

This is a method for the class regsdml. It computes two-sided confidence intervals for testing the two-sided component-wise null hypotheses that tests if a component equals zero with the (approximate) asymptotic Gaussian distribution of the coefficient estimator. The method can be applied to objects of class regsdml, which typically result from a function call to regsdml.

Usage

## S3 method for class 'regsdml'
confint(object,
    parm = NULL,
    level = 0.95,
    print_regsDML = NULL,
    print_safety = NULL,
    print_DML = NULL,
    print_regDML = NULL,
    print_regDML_all_gamma = !is.null(parm),
    print_gamma = FALSE, ...)

Arguments

Value

Confidence intervals for the methods regsDML, the safety device, DML, regDML with the optimal choice of \gamma (including the factor a_N), and regDML with prespecified \gamma-values are returned by setting the respective arguments.

If none of the printing arguments are set, only the results of regsDML are returned if they are available. If they are not available and none of the printing arguments are set, the results from all available methods are returned. If print_regsDML = FALSE, only the results from those methods are returned that are explicitly specified by the printing arguments.

See Also

regsdml, summary.regsdml, coef.regsdml, vcov.regsdml print.regsdml

Examples

## See example(regsdml) for examples.

dmlalg: double machine learning algorithms

Description

The dmlalg package contains implementations of double machine learning (DML) algorithms in R.

Partially linear models with confounding variables

Our goal is to perform inference for the linear parameter in partially linear models with confounding variables. The standard DML estimator of the linear parameter has a two-stage least squares interpretation, which can lead to a large variance and overwide confidence intervals. We apply regularization to reduce the variance of the estimator, which produces narrower confidence intervals that are approximately valid. Nuisance terms can be flexibly estimated with machine learning algorithms.

regsdml

Estimates the linear parameter in a partially linear model with confounding variables with regularized and standard DML methods.

summary.regsdml

A summary method for objects fitted with regsdml.

confint.regsdml

A confint method for objects fitted with regsdml.

coef.regsdml

A coef method for objects fitted with regsdml.

vcov.regsdml

A vcov method for objects fitted with regsdml.

print.regsdml

A print method for objects fitted with regsdml.

Partially linear mixed-effects models with repeated measurements

Our goal is to estimate and perform inference for the linear coefficient in a partially linear mixed-effects model with DML. Machine learning algorithms allows us to incorporate more complex interaction structures and high-dimensional variables.

mmdml

Estimates the linear parameter in a PLMM with repeated measurements using double machine learning.

confint.mmdml

A confint method for objects fitted with mmdml.

fixef.mmdml

A fixef method for objects fitted with mmdml.

print.mmdml

A print method for objects fitted with mmdml.

ranef.mmdml

A ranef method for objects fitted with mmdml.

residuals.mmdml

A residuals method for objects fitted with mmdml.

sigma.mmdml

A sigma method for objects fitted with mmdml.

summary.mmdml

A summary method for objects fitted with mmdml.

vcov.mmdml

A vcov method for objects fitted with mmdml.

VarCorr.mmdml

A VarCorr method for objects fitted with mmdml.

References

C. Emmenegger and P. Bühlmann. Regularizing Double Machine Learning in Partially Linear Endogenous Models, 2021. Preprint arXiv:2101.12525.

C. Emmenegger and P. Bühlmann. Double Machine Learning for Partially Linear Mixed-Effects Models with Repeated Measurements. Preprint arXiv:2108.13657.

Generate data from partially linear mixed-effects model

Description

Generate data from a partially linear mixed-effects model with one or two fixed effects, 2 random effects, and 3 nonparametric variables. The true underlying function of the nonparametric variables are step functions. The random effects and error terms are from a Gaussian distribution.

Usage

example_data_mmdml(beta0, N = 10L, n = 5L)

Arguments

Value

A data frame with the columns resp (the response), id and cask (random effects), w1, w2, and w3 (nonparametric confounders), and x1 if beta0 is of length 1 and x1 and x2 if beta0 is of length 2. The random effects are modelled with "(1|id) + (1|cask:id)".

See Also

mmdml

Examples

## See example(mmdml) for examples

Extract Components from 'mmdml' Fits Imported from 'lme4'

Description

Methods for the class mmdml for generics from lme4.

Usage

fixef(object, ...)
## S3 method for class 'mmdml'
fixef(object, ...)

ranef(object, ...)
## S3 method for class 'mmdml'
ranef(object, ...)

VarCorr(x, sigma = 1, ...)
## S3 method for class 'mmdml'
VarCorr(x, ...)

vcov(object, ...)
## S3 method for class 'mmdml'
vcov(object, ...)

Arguments

Details

fixef.mmdml: Extracts the estimator of the linear coefficient \beta_0, which is a named and numeric vector.

ranef.mmdml: Extracts the random_eff entry from object.

VarCorr.mmdml: The variance and correlation components are computed with the sigma and the theta entries of x as in lmer. For each of the S repetitions, sigma and theta computed on the K sample splits are aggregated by taking the mean. Then, the S mean-aggregated estimates are aggregated by the median. The variance and correlation components are computed with these median-aggregated estimates.

vcov.mmdml: It returns the variance-covariance matrix of the estimator of the linear coefficient is extracted. It is computed based on the asymptotic Gaussian distribution of the estimator. First, for each of the S repetitions, the variance-covariance matrices computed on the K sample splits are aggregated by taking the mean. Second, the S mean-aggregated estimates are aggregated by adding a term correcting for the randomness in the sample splits and by taking the median of these corrected terms. This final corrected and median-aggregated matrix is returned.

Value

See lmer from package lme4.

See Also

mmdml

Examples

## See example(mmdml) for examples

Estimating linear coefficients in partially linear mixed-effects models with repeated measurements using double machine learning.

Description

Our goal is to perform inference for the linear parameter in partially linear mixed-effects models (PLMMs) with repeated measurements using double machine learning (DML).

The function mmdml estimates the linear parameter \beta_0 in the PLMM

Y_i = X_i\beta_0 + g(W_i) + Z_ib_i + \epsilon_{Y_i}, (i = 1, \ldots, N)

for the continuous response Y_i with linear covariates X_i, nonlinear covariates W_i, unobserved random effects b_i, and the error term \epsilon_{Y_i}. For each i, there are n_i repeated observations available. That is, Y_i is an n_i-dimensional vector. The matrix Z_i is fixed. The random effects b_i and the error terms \epsilon_{Y_i} are Gaussian distributed, independent, and independent of b_j and \epsilon_{Y_j}, respectively, for i\neq j. The linear and nonlineare covariates X_i and W_i are random and independent of all random effects and error terms.

The linear parameter \beta_0 can be estimated with a linear mixed-effects modeling approach with maximum likelihood after regressing out W_i nonparametrically from Y_i and X_i using machine learning algorithms. A linear mixed-effects model is estimated on the partialled-out data

Y_i - E[Y_i|W_i] = (X_i - E[X_i|W_i])\beta_0 + Z_ib_i + \epsilon_{Y_i}.

The conditional expectations are estimated with machine learning algorithms and sample splitting, and cross-fitting is used to regain full efficiency of the estimator of beta_0. This estimator is asymptotically Gaussian distributed and efficient.

Usage

mmdml(
  w, x, y, z, data = NULL,
  z_formula = NULL, group = "group",
  K = 2L, S = 100L,
  cond_method = rep("forest", 2),
  params = NULL,
  parallel = c("no", "multicore", "snow"),
  ncpus = 1L, cl = NULL,
  nr_random_eff = if (S > 5) 1L else S,
  nr_res = nr_random_eff
)

Arguments

Details

The estimator of \beta_0 is computed using sample splitting and cross-fitting. The subject-specific data (over i = 1, \ldots, N) is split into K sets that are equally large if possible. For each such set, the nuisance parameters (that is, the conditional expectations E[Y_i|W_i] and E[X_i|W_i]) are estimated on its complement and evaluated on the set itself. Estimators of \beta_0 and the variance parameters are computed for each of the K data sets and are then averaged. If K = 1, no sample splitting is performed. In this case, the nuisance parameters are estimated and predicted on the full sample.

The whole estimation procedure is repeated S times to account for the randomness introduced by the random sample splits. The S repetitions can be run in parallel by specifying the arguments parallel and ncpus. The S estimators of \beta_0 and the variance components are aggregated by taking the median of them. The S variance-covariance matrices of the estimator of \beta_0 are aggregated by first adding a correction term to them that accounts for the random splitting and by afterwards taking the median of the corrected variance-covariance matrices. If d > 1, it can happen that this final matrix is not positive definite anymore, in which case the mean is considered instead. Estimates of the conditional random effects and the residuals are computed in each of the S repetitions. A total number of nr_random_eff and nr_res of them, respectively, is returned. Additionally, the random effects estimates from all S repetitions are aggregated using the mean and returned.

If the design in at least 0.5 * S of the S repetitions is singular, an error message is displayed. If the designs in some but less than 0.5 * S of the S repetitions are singular, another S repetitions are performed. If, in total, at least S repetitions result in a nonsingular design, the results are returned together with a warning message.

The default options of the "spline", "forest", "ols", "lasso", "ridge", and "elasticnet" methods are as follows. With the "spline" method, the function bs from the package splines is employed with degree = 3 and df = ceiling(N ^ (1 / 5)) + 2 if N satisfies (df + 1) * v + 1 > N, where v denotes the number of columns of w and N denotes the sample size. Otherwise, df is consecutively reduced by 1 until this condition is satisfied. The splines are fitted and predicted on different data sets. If they are extrapolated, a warning message is displayed. With the "forest" method, the function randomForest from the package randomForest is employed with nodesize = 5, ntree = 500, na.action = na.omit, and replace = TRUE. With the "ols" method, the default arguments are used and no additional arguments are specified. With the "lasso" and "ridge" methods, the function cv.glmnet from the package glmnet performs 10-fold cross validation by default (argument nfolds) to find the one-standard-error-rule \lambda-parameter. With the "elasticnet" method, the function cv.glmnet from the package glmnet performs 10-fold cross validation (argument nfolds) with alpha = 0.5 by default to find the one-standard-error-rule \lambda-parameter. All default values of the mentioned parameters can be adapted by specifying the argument params.

There are three possibilities to set the argument parallel, namely "no" for serial evaluation (default), "multicore" for parallel evaluation using forking, and "snow" for parallel evaluation using a parallel socket cluster. It is recommended to select RNGkind ("L'Ecuyer-CMRG") and to set a seed to ensure that the parallel computing of the package dmlalg is reproducible. This ensures that each processor receives a different substream of the pseudo random number generator stream. Thus, the results reproducible if the arguments remain unchanged. There is an optional argument cl to specify a custom cluster if parallel = "snow".

The response y needs to be continuous. The covariate w may contain factor variables in its columns. If the variable x contains factor variables, the factors should not be included as factor columns of x. Instead, dummy encoding should be used for all individual levels of the factor. That is, a factor with 4 levels should be encoded with 4 columns where each column consists of 1 and 0 entries indicating the presence of the respective level of the factor.

There are confint, fixef, print, ranef, residuals, sigma, summary, vcov, and VarCorr methods available for objects fitted with mmdml. They are called confint.mmdml, fixef.mmdml, print.mmdml, ranef.mmdml, residuals.mmdml, sigma.mmdml, summary.mmdml, vcov.mmdml, and VarCorr.mmdml, respectively.

Value

A list similar to the output of lmer from package lme4 containing the following entries.

The other elements ngrps, nobs, fitMsgs, cnms, nc, nms, useSc, optinfo, and methTitle are as in lmer. The gradient and Hessian information of optinfo is computed by aggregating the respective information over the S repetitions with the median.

References

C. Emmenegger and P. Bühlmann. Double Machine Learning for Partially Linear Mixed-Effects Models with Repeated Measurements. Preprint arXiv:2108.13657.

See Also

confint, fixef, print, ranef, residuals, sigma, summary, vcov, VarCorr

Examples

## generate data
RNGkind("L'Ecuyer-CMRG")
set.seed(19)
data1 <- example_data_mmdml(beta0 = 0.2)
data2 <- example_data_mmdml(beta0 = c(0.2, 0.2))

## fit models
## Caveat: Warning messages are displayed because the small number of
## observations results in a singular random effects model
fit1 <-
  mmdml(w = c("w1", "w2", "w3"), x = "x1", y = "resp", z = c("id", "cask"),
        data = data1, z_formula = "(1|id) + (1|cask:id)", group = "id", S = 3)

fit2 <-
  mmdml(w = c("w1", "w2", "w3"), x = c("x1", "x2"), y = "resp", z = c("id", "cask"),
        data = data2, z_formula = "(1|id) + (1|cask:id)", group = "id", S = 3)

## apply methods
confint(fit2)
fixef(fit2)
print(fit2)
ranef(fit2)
residuals(fit2)
sigma(fit2)
summary(fit2)
vcov(fit2)
VarCorr(fit2)

Printing mmdml fits

Description

This is a method for the class mmdml. It prints objects of class mmdml that typically result from a function call to mmdml.

Usage

## S3 method for class 'mmdml'
print(x, digits = max(3, getOption("digits") - 3),
    ranef.comp = "Std.Dev.", ...)

Arguments

Value

See lmer.

See Also

mmdml

Examples

## See example(mmdml) for examples

Printing regsdml fits

Description

This is a method for the class regsdml. It prints objects of class regsdml, which typically result from a function call to regsdml.

Usage

## S3 method for class 'regsdml'
print(x, ...)

Arguments

Value

By default, summary(x) is called. Please see summary.regsdml for further details.

See Also

regsdml, summary.regsdml, confint.regsdml, coef.regsdml, vcov.regsdml

Examples

## Generate some data:
set.seed(19)
# true linear parameter
beta0 <- 1
n <- 40
# observed confounder
w <- pi * runif(n, -1, 1)
# instrument
a <- 3 * tanh(2 * w) + rnorm(n, 0, 1)
# unobserved confounder
h <- 2 * sin(w) + rnorm(n, 0, 1)
# linear covariate
x <- -1 * abs(a) - h - 2 * tanh(w) + rnorm(n, 0, 1)
# response
y <- beta0 * x - 3 * cos(pi * 0.25 * h) + 0.5 * w ^ 2 + rnorm(n, 0, 1)

## Estimate the linear coefficient from x to y
## (The parameters are chosen small enough to make estimation fast):
## Caveat: A spline estimator is extrapolated, which raises a warning message.
## Extrapolation lies in the nature of our method. To omit the warning message
## resulting from the spline estimator, another estimator may be used.
fit <- regsdml(a, w, x, y,
               gamma = exp(seq(-4, 1, length.out = 4)),
               S = 3,
               do_regDML_all_gamma = TRUE,
               cond_method = c("forest",  # for E[A|W]
                               "spline",  # for E[X|W]
                               "spline"), # for E[Y|W]
               params = list(list(ntree = 1), NULL, NULL))
print(fit)

Estimating linear coefficients with double machine learning (DML)

Description

Our goal is to perform inference for the linear parameter in partially linear models with confounding variables. The standard double machine learning (DML) estimator of the linear parameter has a two-stage least squares interpretation, which can lead to a large variance and overwide confidence intervals. We apply regularization to reduce the variance of the estimator, which produces narrower confidence intervals that remain approximately valid.

The function regsdml estimates the linear parameter \beta_0 in the partially linear model

Y = X^T\beta_0 + g(W) + h(H) + \epsilon_Y

of the continuous response Y with linear covariates X, nonlinear covariates W, unobserved confounding variables H, and the error term \epsilon_Y. An additional variable A is required that is not part of the right-hand side defining Y. The variable A acts as an instrument after W is regressed out of it.

The linear parameter \beta_0 can be estimated with a two-stage least squares (TSLS) approach ("standard" DML) or with regularized approaches (regDML, regsDML). All approaches use double machine learning. The TSLS approach regresses the residual Y - E[Y|W] on X - E[X|W] using the instrument A - E[A|W]. The regularized approaches minimize an objective function that equals \gamma times the objective function of TSLS plus an objective function that partials out A - E[A|W] from the residual quantity Y - E[Y|W] - (X - E[X|W])^T\beta. The different regularization approaches choose different regularization parameters \gamma. The conditional expectations act as nuisance parameters and are estimated with machine learning algorithms. All approaches use sample splitting and cross-fitting to estimate \beta_0.

Usage

regsdml(
  a, w, x, y, data = NULL,
  DML = c("DML2", "DML1"),
  K = 2L,
  gamma = exp(seq(-4, 10, length.out = 100)),
  aN = NULL,
  do_regsDML = TRUE,
  do_safety = FALSE,
  do_DML = do_regDML || do_regsDML || do_safety,
  do_regDML = FALSE,
  do_regDML_all_gamma = FALSE,
  safety_factor = 0.7,
  cond_method = rep("spline", 3),
  params = NULL,
  level = 0.95,
  S = 100L,
  parallel = c("no", "multicore", "snow"),
  ncpus = 1L,
  cl = NULL
)

Arguments

Details

The estimator of \beta_0 is computed using sample splitting and cross-fitting. Irrespective of which methods are performed, the data is split into K sets that are equally large if possible. For each such set, the nuisance parameters (that is, the conditional expectations E[A|W], E[X|W], and E[Y|W]) are estimated on its complement and evaluated on the set itself. If DML = "DML1", then K individual estimators are computed for each of the K data sets and are then averaged. If DML = "DML2", the nuisance parameter matrices are first assembled before the estimator of \beta_0 is computed. This enhances stability of the coefficient estimator compared to "DML1". If K = 1, no sample splitting is performed. In this case, the nuisance parameters are estimated and predicted on the full sample.

The whole estimation procedure can be repeated S times to account for the randomness introduced by the random sample splits. The S repetitions can be run in parallel by specifying the arguments parallel and ncpus. The S estimators of \beta_0 are aggregated by taking the median of them. The S variance-covariance matrices are aggregated by first adding a correction term to them that accounts for the random splitting and by afterwards taking the median of the corrected variance-covariance matrices. If d > 1, it can happen that this final matrix is not positive definite anymore, in which case the mean is considered instead.

If the design in at least 0.5 * S of the S repetitions is singular, an error message is displayed. If the designs in some but less than 0.5 * S of the S repetitions are singular, another S repetitions are performed. If, in total, at least S repetitions result in a nonsingular design, the results are returned together with a warning message.

The regularized estimators and their associated mean squared errors (MSEs) are computed for the regularization parameters \gamma of the grid gamma. These estimators are returned if the argument do_regDML_all_gamma is set to TRUE. The \gamma-value whose corresponding regularized estimator from the do_regDML_all_gamma method achieves the smallest MSE is multiplied by aN, leading to \gamma'. The do_regDML_all_gamma estimator with regularization parameter \gamma' is called regDML. The regsDML estimator equals regDML or DML depending on whose variance is smaller. If \beta_0 is of larger dimension than 1, the MSE computations and the variance comparison step are performed with the sum of the diagonal entries of the respective variance-covariance matrices.

If do_safety = TRUE, a \gamma value is chosen such that the regularized estimator among do_regDML_all_gamma with this value of \gamma has a variance that is just not smaller than safety_factor times the variance of DML. If \beta_0 is of larger dimension than 1, the sum of the diagonal entries of the respective variance-covariance matrices is taken as a measure of variance. If the regularization scheme leads to considerable variance reductions, it is possible that this safety device cannot be applied. In this case, a respective message is returned.

The default options of the "spline", "forest", "ols", "lasso", "ridge", and "elasticnet" methods are as follows. With the "spline" method, the function bs from the package splines is employed with degree = 3 and df = ceiling(N ^ (1 / 5)) + 2 if N satisfies (df + 1) * v + 1 > N, where v denotes the number of columns of w and N denotes the sample size. Otherwise, df is consecutively reduced by 1 until this condition is satisfied. The splines are fitted and predicted on different data sets. If they are extrapolated, a warning message is displayed. With the "forest" method, the function randomForest from the package randomForest is employed with nodesize = 5, ntree = 500, na.action = na.omit, and replace = TRUE. With the "ols" method, the default arguments are used and no additional arguments are specified. With the "lasso" and "ridge" methods, the function cv.glmnet from the package glmnet performs 10-fold cross validation by default (argument nfolds) to find the one-standard-error-rule \lambda-parameter. With the "elasticnet" method, the function cv.glmnet from the package glmnet performs 10-fold cross validation (argument nfolds) with alpha = 0.5 by default to find the one-standard-error-rule \lambda-parameter. All default values of the mentioned parameters can be adapted by specifying the argument params.

There are three possibilities to set the argument parallel, namely "no" for serial evaluation (default), "multicore" for parallel evaluation using forking, and "snow" for parallel evaluation using a parallel socket cluster. It is recommended to select RNGkind ("L'Ecuyer-CMRG") and to set a seed to ensure that the parallel computing of the package dmlalg is reproducible. This ensures that each processor receives a different substream of the pseudo random number generator stream. Thus, the results reproducible if the arguments remain unchanged. There is an optional argument cl to specify a custom cluster if parallel = "snow".

The response y needs to be continuous. The covariate w may contain factor variables in its columns. If the variables a and x contain factor variables, the factors should not be included as factor columns of a or x. Instead, dummy encoding should be used for all individual levels of the factor. That is, a factor with 4 levels should be encoded with 4 columns where each column consists of 1 and 0 entries indicating the presence of the respective level of the factor.

There are summary, confint, coef, vcov, and print methods available for objects fitted with regsdml. They are called summary.regsdml, confint.regsdml, coef.regsdml, vcov.regsdml, and print.regsdml, respectively.

Value

A list containing some of the lists regsDML_statistics, regDML_safety_statistics, DML_statistics, regDML_statistics, and regDML_all_gamma_statistics is returned. The individual sublists contain the following arguments supplemented by an additional suffix specifying the method they correspond to.

The list regsDML_statistics contains the following additional entries:

If the safety device is applicable, the list regDML_safety_statistics contains the following additional entries:

If the safety device is not applicable, the list regDML_safety_statistics contains message_safety as its only entry.

The list regDML_statistics contains the following additional entry:

The list regDML_all_gamma_statistics is a list of the same length as the grid gamma, where each individual list is of the structure just described.

References

C. Emmenegger and P. Bühlmann. Regularizing Double Machine Learning in Partially Linear Endogenous Models, 2021. Preprint arXiv:2101.12525.

See Also

summary.regsdml, confint.regsdml, coef.regsdml, vcov.regsdml print.regsdml

Examples

## Generate some data:
RNGkind("L'Ecuyer-CMRG")
set.seed(19)
# true linear parameter
beta0 <- 1
n <- 40
# observed confounder
w <- pi * runif(n, -1, 1)
# instrument
a <- 3 * tanh(2 * w) + rnorm(n, 0, 1)
# unobserved confounder
h <- 2 * sin(w) + rnorm(n, 0, 1)
# linear covariate
x <- -1 * abs(a) - h - 2 * tanh(w) + rnorm(n, 0, 1)
# response
y <- beta0 * x - 3 * cos(pi * 0.25 * h) + 0.5 * w ^ 2 + rnorm(n, 0, 1)

## Estimate the linear coefficient from x to y
## (The parameters are chosen small enough to make estimation fast):
## Caveat: A spline estimator is extrapolated, which raises a warning message.
## Extrapolation lies in the nature of our method. To omit the warning message
## resulting from the spline estimator, another estimator may be used.
fit <- regsdml(a, w, x, y,
               gamma = exp(seq(-4, 1, length.out = 4)),
               S = 3,
               do_regDML_all_gamma = TRUE,
               cond_method = c("forest",  # for E[A|W]
                               "spline",  # for E[X|W]
                               "spline"), # for E[Y|W]
               params = list(list(ntree = 1), NULL, NULL))
## parm = c(2, 3) prints an additional summary for the 2nd and 3rd gamma-values
summary(fit, parm = c(2, 3),
        correlation = TRUE,
        print_gamma = TRUE)
confint(fit, parm = c(2, 3),
        print_gamma = TRUE)
coef(fit) # coefficients
vcov(fit) # variance-covariance matrices

## Alternatively, provide the data in a single data frame
## (see also caveat above):
data <- data.frame(a = a, w = w, x = x, y = y)
fit <- regsdml(a = "a", w = "w", x = "x", y = "y", data = data,
               gamma = exp(seq(-4, 1, length.out = 4)),
               S = 3)

## With more realistic parameter choices:
if (FALSE) {
  fit <- regsdml(a, w, x, y,
                 cond_method = c("forest",  # for E[A|W]
                                 "spline",  # for E[X|W]
                                 "spline")) # for E[Y|W]
  summary(fit)
  confint(fit)

  ## Alternatively, provide the data in a single data frame:
  ## (see also caveat above):
  data <- data.frame(a = a, w = w, x = x, y = y)
  fit <- regsdml(a = "a", w = "w", x = "x", y = "y", data = data)
}

Confidence Intervals for coefficient estimates of regsDML fits

Description

A list whose elements correspond to the potentially scaled first nr_res sets of residuals of the S residuals.

Usage

## S3 method for class 'mmdml'
residuals(object, scaled = FALSE, ...)

Arguments

Value

A list whose elements correspond to the first nr_res sets of residuals of the S residuals.

See Also

mmdml

Examples

## See example(mmdml) for examples

Extract Residual Standard Deviation 'Sigma' from mmdml Fits

Description

Extract the estimated standard deviation of the errors, the “residual standard deviation”, from a fitted mmdml model.

Usage

## S3 method for class 'mmdml'
sigma(object, ...)

Arguments

Value

A number representing the estimated standard deviation. First, for each of the S repetitions, the standard deviations computed on the K sample splits are aggregated by taking the mean. Second, the S mean-aggregated estimates are aggregated by the median. This final value is returned.

See Also

mmdml

Examples

## See example(mmdml) for examples

Summarizing mmdml fits

Description

This is a method for the class mmdml. It summarizes objects of class mmdml that typically result from a function call to mmdml.

Usage

## S3 method for class 'mmdml'
summary(object,
    correlation = (p <= getOption("lme4.summary.cor.max")),
    nr_res = NULL, ...)

Arguments

Value

Prints a summary output similar to lmer from package lme4.

See Also

mmdml

Examples

## See example(mmdml) for examples

Summarizing regsdml fits

Description

This is a method for the class regsdml. It summarizes objects of class regsdml, which typically result from a function call to regsdml.

Usage

## S3 method for class 'regsdml'
summary(object,
    print_regsDML = NULL,
    print_safety = NULL,
    print_DML = NULL,
    print_regDML = NULL,
    print_regDML_all_gamma = !is.null(parm),
    parm = NULL,
    correlation = FALSE,
    print_gamma = FALSE, ...)

Arguments

Value

Summary statistics of the methods regsDML, the safety device, DML, regDML with the optimal choice of \gamma (including the factor a_N), and regDML with prespecified \gamma-values are returned by setting the respective arguments. It is possible to return the respective gamma-values and variance-covariance matrices.

If none of the printing arguments are set, only the results of regsDML are returned if they are available. If they are not available and none of the printing arguments are set, the results from all available methods are returned. If print_regsDML = FALSE, only the results from those methods are returned that are explicitly specified by the printing arguments.

See Also

regsdml, confint.regsdml, coef.regsdml, vcov.regsdml print.regsdml

Examples

## See example(regsdml) for examples

Accessing the variance-covariance matrices of regsdml fits

Description

This is a method for the class regsdml. It returns the variance-covariance matrices of the coefficients from objects of class regsdml, which typically result from a function call to regsdml.

Usage

## S3 method for class 'regsdml'
vcov(object,
    print_regsDML = NULL,
    print_safety = NULL,
    print_DML = NULL,
    print_regDML = NULL,
    print_regDML_all_gamma = !is.null(parm),
    parm = NULL,
    print_gamma = FALSE, ...)

Arguments

Value

Variance-covariance matrices of the methods regsDML, the safety device, DML, regDML with the optimal choice of \gamma (including the factor a_N), and regDML with prespecified \gamma-values are returned by setting the respective arguments. It is possible to return the respective gamma-values.

If none of the printing arguments are set, only the results of regsDML are returned if they are available. If they are not available and none of the printing arguments are set, the results from all available methods are returned. If print_regsDML = FALSE, only the results from those methods are returned that are explicitly specified by the printing arguments.

See Also

regsdml, summary.regsdml, confint.regsdml, coef.regsdml print.regsdml

Examples

## See example(regsdml) for examples